Complex Roots of Unity include 1

Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity.

Then $1 \in U_n$.

That is, $1$ is always one of the complex $n$th roots of unity of any $n$.

Proof

By definition of integer power:

$1^n = 1$

for all $n$.

Hence the result.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity